Algebra Help Calculators Expression Factoring
Two hard algebra 2 word problems- HELP! One factoring problem.?
1) The volume in cubic feet of a box can be expressed as v(x)= x^3-7x^2+12x.
The width of the box is (3-x)
a. factor the polynomial to find linear expressions for the height and length.
b. describe a realistic domain for the function
c. find the max. volume of the box, round to the thousandths place.
d. what is the volume of the box when x=7
2) The formula for the volume of a rectangular prism with width w is V= (3-2w)(w)(7-2w)
a. find roots
b. what is the reasonable domain of the volume?
c. what is the max. volume and what is the width it occurs at?
d. put the formula for the volume of the rectangular prism in standard form.
3) 64x^3 + 8
(I tried taking the cubes of all the numbers but it didn’t come out right in my calculator)
Here’s what I did: (4x+2)^3 since 4 is the cube of 64 and 2 is the cube of 8. :/
how did you find the roots?
3) sum of cubes is (a^3 + b^3) = (a + b)(a^2 – ab + b^2)
so 64x^3 + 8 = (4x + 2)(16x^2 – 8x + 4)
factor further to give: 2(x + 1)(4(4x^2 – 2x + 1) = 8(x + 1)(4x^2 – 2x + 1)
however, you should always start with the GCF, so the first step should be:
8(8x^3 + 1) = 8(2x + 1)(4x^2 – 2x + 1) more directly!
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2)
a) roots: w = 3/2 , w = 0 , w = 7/2
edit: the roots are the solutions to v(w) = 0
in factored form, that means either 3 – 2w = 0 ==> w = 3/2
or w = 0, or 7 – 2w = 0 ==> w = 7/2
b)we need all dimensions to be positive
so w > 0, and 3 – 2w > 0 ==> w < 3/2
domain: (0 , 3/2)
c) without calculus, graph V and find the maximum in the window (0 , 3/2)
d) (3 - 2w)(7 - 2w) = 21 - 20w + 4w^2
w(21 - 20w + 4w^2) = 4w^3 - 20w^2 + 21w
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1)
a) x^3 - 7x^2 + 12x = x(x^2 - 7x + 12) = x(x - 4)(x - 3)
however, if the width is 3 - x, then length = 4 - x
height = x
v(x) = x(4 - x)(3 - x) (both of the binomials were multiplied by -1 from the original factoring, and -1 * -1 = 1, so the volume is unchanged)
b) all dimensions must be positive, so x > 0 and x < 3
domain: (0 , 3)
c) graph this and find the max in the window (0 , 3)
d) plug in x = 7: v(7) = 7(3 - 7)(4 - 7) = 7(-4)(-3)= 84
Pre-Calculus: Trig Equations and Quadratic Formula
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